Problem 29 - Chemistry: An Atoms-Focused Approach

Question

What is the length of an edge of the unit cell when barium (atomic radius \(222 \mathrm{pm}\) ) crystallizes in a crystal lattice of bcc unit cells?

Step-by-Step Solution

Answer

Answer: The length of an edge of the barium unit cell in a body-centered cubic lattice is approximately 509.4 pm.

Step 1: Recall the body-centered cubic (bcc) structure

In a bcc unit cell, there is an atom at each corner of a cube and one atom in the center. So, there are 8 corner atoms and 1 center atom. However, each corner atom is shared by 8 neighboring cells, so in total, there are 2 atoms per unit cell (1/8 from each corner atom and 1 from the center atom).

Step 2: Use the relationship between the atomic radius (r) and the edge length (a) in a bcc lattice

In a bcc structure, the relationship between the atomic radius (r) and the edge length (a) is given by the formula: \[a = 4r \sqrt{3}/3\] Here, we are given the atomic radius of barium as 222 pm. We will plug this value into the formula to find the edge length.

Step 3: Calculate the edge length (a)

Substitute the atomic radius of barium (r = 222 pm) into the formula: \[a = 4(222\,\mathrm{pm}) \frac{\sqrt{3}}{3}\] Now, we can calculate the edge length (a): \[a \approx 509.4\,\mathrm{pm}\] Hence, the length of an edge of the barium unit cell when it crystallizes in a bcc lattice is approximately 509.4 pm.